The Liouville equation is well known to be linearizable by a point transformation. It has an infinite dimensional Lie point symmetry algebra isomorphic to a direct sum of two Virasoro algebras. We show that it is not possible to discretize the equation keeping the entire symmetry algebra as point symmetries. We do however construct a difference system approximating the Liouville equation that is invariant under the maximal finite subgroup ##IMG## [http://ej.iop.org/images/1751-8121/48/2/025204/jpa504755ieqn1.gif] {$S{{L}_{x}}(2,mathbb{R})otimes S{{L}_{y}}(2,mathbb{R})$} . The invariant scheme is an explicit one and provides a much better approximation of exact solutions than a comparable standard (noninvariant) scheme and also than a scheme invariant under an infinite dimensional group of generalized symmetries.
Lie-point symmetries of the discrete Liouville equation
MARTINA, Luigi;
2015-01-01
Abstract
The Liouville equation is well known to be linearizable by a point transformation. It has an infinite dimensional Lie point symmetry algebra isomorphic to a direct sum of two Virasoro algebras. We show that it is not possible to discretize the equation keeping the entire symmetry algebra as point symmetries. We do however construct a difference system approximating the Liouville equation that is invariant under the maximal finite subgroup ##IMG## [http://ej.iop.org/images/1751-8121/48/2/025204/jpa504755ieqn1.gif] {$S{{L}_{x}}(2,mathbb{R})otimes S{{L}_{y}}(2,mathbb{R})$} . The invariant scheme is an explicit one and provides a much better approximation of exact solutions than a comparable standard (noninvariant) scheme and also than a scheme invariant under an infinite dimensional group of generalized symmetries.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
